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main.go
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package main
import (
"flag"
"fmt"
"math/big"
"time"
"github.com/fedejinich/lattigo/v5/ring"
"github.com/fedejinich/lattigo/v5/utils"
)
// Vectorized oblivious evaluation is a two-party protocol for the function f(x) = ax + b where a sender
// inputs the field elements a, b, and a receiver inputs x and learns f(x).
//
// This example implements the secret-key and passively-secure vOLE protocol of Figure 5 of
// "Efficient Protocols for Oblivious Linear Function Evaluation from Ring-LWE" by Baum C. et al.
// https://eprint.iacr.org/2020/970
//
//
// We work in the ring R = Z_{Q}/(X^N + 1), for Q the product of k NTT-friendly primes and N a power of two.
// We define Q > P > M with P|Q and M|Q and M|P, i.e. P and M are themselves factors of Q.
// Thus, we also have R_{M} -> R_{P} -> R_{Q}. A polynomial mod M implies that this polynomial belongs to R_{M};
// the same follows for mod P and mod Q.
//
// Given two parties Alice and Bob, with their respective secret inputs v and u (mod M), the goal of this protocol is
// for Alice and Bob to construct the values alpha, beta (mod M) such that alpha + beta = u*v (mod M).
//
// 1. Setup phase
// (a) Alice and Bob sample each a low-norm secret polynomial: skAlice and skBob (mod Q) respectively and set
// sigmaAlice + sigmaBob = skAlice * skBob (mod Q)
// (b) Alice and Bob sample two public random polynomials: a and a' (mod Q)
// 2. First Message. Given u (mod M), Bob's input
// (a) Bob samples a noise polynomial eBob (mod Q) from a small-variance truncated discrete Gaussian distribution
// and sends c = (Q/P) * u + a*skBob + eBob to Alice (mod P)
// (b) Alice computes rhoAlice = (skAlice * c - a * sigmaAlice) * (P/Q) (mod P)
// (c) Bob computes rhoBob = -(a * sigmaBob) * (P/Q) (mod P)
//
// At this step, it should hold that u * skAlice = rhoAlice + rhoBob (mod P)
//
// 3. Second Message. Given v, Alice's input
// (a) Alice samples a noise polynomial eAlice (mod P) from a small-variance truncated discrete Gaussian distribution
// and sends d = (P/M) * v + (a'*skAlice + eAlice) (mod P) to Bob
// (b) Bob outputs beta = (u * d - a' * rhoBob) * (M/P) (mod M)
// (c) Alice outputs alpha = -(a' * rhoAlice) * (M/P) (mod M)
//
// It should hold that u * v = alpha + beta (mod M)
var flagShort = flag.Bool("short", false, "run the example on the 3 first parameters sets only.")
type parameters struct {
logN int // Ring degree
logQ [2]int // logQ[0] = #Primes, logQ[1] = Primes bit-size
n int // Number of vOLE to run
plevel int // Maximum level of the modulus P (level 0 is the lowest)
mlevel int // Maximum level of the modulus M (level 0 is the lowest)
}
var benchParameters = []parameters{
{logN: 13, logQ: [2]int{4, 60}, n: 1, plevel: 2, mlevel: 0},
{logN: 14, logQ: [2]int{6, 60}, n: 1, plevel: 3, mlevel: 0},
{logN: 14, logQ: [2]int{8, 60}, n: 1, plevel: 5, mlevel: 1},
{logN: 13, logQ: [2]int{4, 60}, n: 8, plevel: 2, mlevel: 0},
{logN: 14, logQ: [2]int{6, 60}, n: 8, plevel: 3, mlevel: 0},
{logN: 14, logQ: [2]int{8, 60}, n: 8, plevel: 5, mlevel: 1},
{logN: 13, logQ: [2]int{4, 60}, n: 64, plevel: 2, mlevel: 0},
{logN: 14, logQ: [2]int{6, 60}, n: 64, plevel: 3, mlevel: 0},
{logN: 14, logQ: [2]int{8, 60}, n: 64, plevel: 5, mlevel: 1},
{logN: 13, logQ: [2]int{4, 60}, n: 128, plevel: 2, mlevel: 0},
{logN: 14, logQ: [2]int{6, 60}, n: 128, plevel: 3, mlevel: 0},
{logN: 14, logQ: [2]int{8, 60}, n: 128, plevel: 5, mlevel: 1},
}
type vOLErings struct {
ringQ *ring.Ring // Z_{Q}/(X^N+1)
qDivP *big.Int // Q/P
pDivM *big.Int // P/M
}
func newvOLErings(params parameters) *vOLErings {
if params.mlevel >= params.plevel {
panic("mlevel must be strictly smaller than plevel")
}
var err error
N := 1 << params.logN
rings := new(vOLErings)
// Generate logQ[0] NTT-friendly primes each close to 2^logQ[1]
primes := ring.GenerateNTTPrimes(params.logQ[1], 2*N, params.logQ[0])
if rings.ringQ, err = ring.NewRing(N, primes); err != nil {
panic(err)
}
rings.qDivP = ring.NewInt(1)
for _, qi := range primes[params.plevel+1:] {
rings.qDivP.Mul(rings.qDivP, ring.NewUint(qi))
}
rings.pDivM = ring.NewInt(1)
for _, qi := range primes[params.mlevel+1 : params.plevel+1] {
rings.pDivM.Mul(rings.pDivM, ring.NewUint(qi))
}
return rings
}
type lowNormSampler struct {
baseRing *ring.Ring
coeffs []*big.Int
}
func newLowNormSampler(baseRing *ring.Ring) (lns *lowNormSampler) {
lns = new(lowNormSampler)
lns.baseRing = baseRing
lns.coeffs = make([]*big.Int, baseRing.N())
return
}
// Samples a uniform polynomial in Z_{norm}/(X^N + 1)
func (lns *lowNormSampler) newPolyLowNorm(norm *big.Int) (pol *ring.Poly) {
pol = lns.baseRing.NewPoly()
for i := range lns.coeffs {
lns.coeffs[i] = ring.RandInt(norm)
}
lns.baseRing.AtLevel(pol.Level()).SetCoefficientsBigint(lns.coeffs, pol)
return
}
func main() {
flag.Parse()
paramsSets := benchParameters
if *flagShort {
paramsSets = paramsSets[:6] // skips the benchmarks for n=64 and 128 in -short mode
}
for _, param := range paramsSets {
// Crypto Context Initialization
volerings := newvOLErings(param)
ringQ := volerings.ringQ
n := param.n
qlevel := ringQ.MaxLevel()
plevel := param.plevel
mlevel := param.mlevel
fmt.Printf("Params : n=%d logN=%d qlevel=%d plevel=%d mlevel=%d\n", n, param.logN, qlevel, plevel, mlevel)
prng, err := utils.NewPRNG()
if err != nil {
panic(err)
}
ternarySamplerMontgomeryQ := ring.NewTernarySampler(prng, ringQ, 1.0/3.0, true)
gaussianSamplerQ := ring.NewGaussianSampler(prng, ringQ, 3.2, 19)
uniformSamplerQ := ring.NewUniformSampler(prng, ringQ)
lowNormUniformQ := newLowNormSampler(ringQ)
var elapsed, TotalTime, AliceTime, BobTime time.Duration
start := time.Now()
// ********* 1. SETUP *********
buff := ringQ.NewPoly()
// NTT(MForm(skBob))
skBob := ternarySamplerMontgomeryQ.ReadNew()
ringQ.NTT(skBob, skBob)
// NTT(MForm(skAlice))
skAlice := ternarySamplerMontgomeryQ.ReadNew()
ringQ.NTT(skAlice, skAlice)
// NTT(MForm(sigmaAlicelice))
sigmaAlice := uniformSamplerQ.ReadNew()
// NTT(MForm(sigmaBobob))
sigmaBob := ringQ.NewPoly()
// NTT(MForm(skBob * skAlice))
ringQ.MulCoeffsMontgomery(skAlice, skBob, sigmaBob)
// NTT(MForm(sigmaBob)) = NTT(MForm(ska_a * skAlice) - MForm(sigmaAlice))
ringQ.Sub(sigmaBob, sigmaAlice, sigmaBob)
a := make([]*ring.Poly, n)
aprime := make([]*ring.Poly, n)
// Sample common random poly vectors
// NTT(a) in Z_Q
// NTT(MForm(a')) in Z_P
for i := range a {
a[i] = uniformSamplerQ.ReadNew()
aprime[i] = uniformSamplerQ.AtLevel(plevel).ReadNew()
ringQ.AtLevel(plevel).MForm(aprime[i], aprime[i])
}
elapsed = time.Since(start)
fmt.Printf("Setup : %s\n", elapsed)
// Generate inputs and allocate memory
start = time.Now()
u := make([]*ring.Poly, n)
v := make([]*ring.Poly, n)
c := make([]*ring.Poly, n)
d := make([]*ring.Poly, n)
rhoAlice := make([]*ring.Poly, n)
rhoBob := make([]*ring.Poly, n)
alpha := make([]*ring.Poly, n)
beta := make([]*ring.Poly, n)
tmp := ringQ.NewPoly()
for i := 0; i < n; i++ {
// Generate uniform inputs u and v (in R_m) in the time domain
u[i] = lowNormUniformQ.newPolyLowNorm(ringQ.ModulusAtLevel[mlevel])
ringQ.NTT(u[i], u[i])
v[i] = lowNormUniformQ.newPolyLowNorm(ringQ.ModulusAtLevel[mlevel])
c[i] = ringQ.NewPoly()
d[i] = ringQ.AtLevel(plevel).NewPoly()
rhoAlice[i] = ringQ.NewPoly()
rhoBob[i] = ringQ.NewPoly()
alpha[i] = ringQ.AtLevel(plevel).NewPoly()
beta[i] = ringQ.AtLevel(plevel).NewPoly()
}
elapsed = time.Since(start)
fmt.Printf("Gen Inputs : %s\n", elapsed)
// ********* 2. FIRST MESSAGE *********
// First Message starts (Bob)
start = time.Now()
for i := 0; i < n; i++ {
// c = e
gaussianSamplerQ.AtLevel(qlevel).ReadAndAdd(c[i])
// c = NTT(e)
ringQ.NTT(c[i], c[i])
// tmp = NTT(u * Q/P)
ringQ.MulScalarBigint(u[i], volerings.qDivP, tmp)
// c = NTT(u * (Q/P) + e)
ringQ.Add(c[i], tmp, c[i])
// c = NTT(u * (Q/P) + e) + NTT(a) * MForm(NTT(skAlice))
// c = NTT(u * (Q/P) + e + a*skAlice)
ringQ.MulCoeffsMontgomeryThenAdd(a[i], skBob, c[i])
}
elapsed = time.Since(start)
fmt.Printf("2.(a) : %s\n", elapsed)
TotalTime = elapsed
BobTime = elapsed
// Alice
start = time.Now()
for i := 0; i < n; i++ {
// rhoAlice = NTT(MForm(skBob)) * NTT(u * (Q/P) + e + a*skAlice)
// rhoAlice = NTT(skBob * (u * (Q/P) + e + a*skAlice))
ringQ.MulCoeffsMontgomery(skAlice, c[i], rhoAlice[i])
// rhoAlice = NTT(skBob * (u * (Q/P) + e + a*skAlice)) - NTT(a) * NTT(MForm(sigmaAlice))
// rhoAlice = NTT(skBob * (u * (Q/P) + e + a*skAlice) - a*sigmaAlice)
ringQ.MulCoeffsMontgomeryThenSub(a[i], sigmaAlice, rhoAlice[i])
// rhoAlice = NTT(skBob * u) + NTT(a*skBob*skAlice - a*sigmaAlice) * (P/Q)
ringQ.AtLevel(qlevel).DivRoundByLastModulusManyNTT(qlevel-plevel, rhoAlice[i], buff, rhoAlice[i])
rhoAlice[i].Resize(plevel)
}
elapsed = time.Since(start)
fmt.Printf("2.(c) : %s\n", elapsed)
TotalTime += elapsed
AliceTime = elapsed
// Bob
start = time.Now()
for i := 0; i < n; i++ {
// rhoBob = NTT(a) * NTT(MForm(sigmaBob))
// rhoBob = NTT(a * sigmaBob)
ringQ.MulCoeffsMontgomery(a[i], sigmaBob, rhoBob[i])
// rhoBob = NTT(a * sigmaBob * (P/Q))
ringQ.AtLevel(qlevel).DivRoundByLastModulusManyNTT(qlevel-plevel, rhoBob[i], buff, rhoBob[i])
rhoBob[i].Resize(plevel)
// rhoBob = NTT(-a * sigmaBob) * (P/Q)
ringQ.AtLevel(plevel).Neg(rhoBob[i], rhoBob[i])
}
elapsed = time.Since(start)
fmt.Printf("2.(d) : %s\n", elapsed)
TotalTime += elapsed
BobTime += elapsed
fmt.Printf("Total time First Message : %s\n", TotalTime)
fmt.Printf("Alice time First Message : %s\n", AliceTime)
fmt.Printf("Bob time First Message : %s\n", BobTime)
// ********* VERIFY CORRECTNESS *********
checkMessage1a := ringQ.AtLevel(plevel).NewPoly()
checkMessage1b := ringQ.AtLevel(plevel).NewPoly()
nerrors := 0
for i := 0; i < n; i++ {
// checkMessage1a = NTT(u) * NTT(MForm(skBob))
ringQ.AtLevel(plevel).MulCoeffsMontgomery(u[i], skAlice, checkMessage1a)
// rhoAlice = NTT(skBob * u) + NTT(a*skBob*skAlice - a*sigmaAlice) * P/Q
// rhoBob = NTT(-a * sigmaBob) * P/Q
// NTT(a*skBob*skAlice - a*sigmaAlice) * (P/Q) = NTT(a*skBob*skAlice) - NTT(a) * NTT(MForm(ska_a * skAlice - sigmaBob)
// = NTT(a*skBob*skAlice - a * (skBob*skAlice - sigmaBob))
// = NTT(a*skBob*skAlice - a*skBob*sk-b + a*sigmaBob)
// = NTT(a*sigmaBob) * P/Q
// -> rhoAlice + rhoBob = NTT(skBob * u) + NTT(-a * sigmaBob) * (P/Q) + NTT(a*sigmaBob) * P/Q
// = NTT(skBob * u)
ringQ.AtLevel(plevel).Add(rhoAlice[i], rhoBob[i], checkMessage1b)
if !ringQ.AtLevel(plevel).Equal(checkMessage1a, checkMessage1b) {
nerrors++
}
}
fmt.Printf("Errors First Message : %d\n", nerrors)
// ********* 3. SECOND MESSAGE *********
// Second Message, Alice
start = time.Now()
ringQ.AtLevel(plevel).IMForm(skAlice, skAlice)
for i := 0; i < n; i++ {
// d = v * P/M
ringQ.AtLevel(plevel).MulScalarBigint(v[i], volerings.pDivM, d[i])
// d = v * (P/M) + e
gaussianSamplerQ.AtLevel(plevel).ReadAndAdd(d[i])
// d = NTT(v * (P/M) + e)
ringQ.AtLevel(plevel).NTT(d[i], d[i])
// d = NTT(v * (P/M) + e) + NTT(MForm(a'))*NTT(skAlice)
// = NTT(v * (P/M) + e + a'*skAlice)
ringQ.AtLevel(plevel).MulCoeffsMontgomeryThenAdd(aprime[i], skAlice, d[i])
}
elapsed = time.Since(start)
fmt.Printf("3.(a) : %s\n", elapsed)
TotalTime = elapsed
AliceTime = elapsed
// Bob
start = time.Now()
for i := 0; i < n; i++ {
// beta = NTT(v * (P/M) + e + a'*skAlice) * NTT(MForm(u))
// = NTT(u * (v * (P/M) + e + a'*skAlice))
ringQ.AtLevel(plevel).MForm(u[i], u[i])
ringQ.AtLevel(plevel).MulCoeffsMontgomery(u[i], d[i], beta[i])
// beta = NTT(u * (v * (P/M) + e + a'*skAlice)) - NTT(MForm(a')) * NTT(-a * sigmaBob) * (P/Q)
// = NTT(u * (v * (P/M) + e + a'*skAlice) - a'*-a*sigmaBob * (P/Q))
ringQ.AtLevel(plevel).MulCoeffsMontgomeryThenSub(aprime[i], rhoBob[i], beta[i])
// beta = u*(v * (P/M) + e + a'*skAlice) * u - a'*-a*sigmaBob * (P/Q)
ringQ.AtLevel(plevel).INTT(beta[i], beta[i])
// beta = (u*(v * (P/M) + e + a'*skAlice) * u - a'*-a*sigmaBob * (P/Q)) * (M/P)
// = (M/P) * u * (v * (P/M) + e + a'*skAlice) - a'*-a*sigmaBob * (M/Q)
// = u*v + a'*skAlice*u*(M/P) + a'*a*sigmaBob * (M/Q)
ringQ.AtLevel(plevel).DivRoundByLastModulusMany(plevel-mlevel, beta[i], buff, beta[i])
beta[i].Resize(mlevel)
}
elapsed = time.Since(start)
fmt.Printf("3.(c) : %s\n", elapsed)
TotalTime += elapsed
BobTime = elapsed
// Alice
start = time.Now()
for i := 0; i < n; i++ {
// alpha = NTT(MForm(a')) * (NTT(skAlice * u) + NTT(a*skBob*skAlice - a*sigmaAlice) * (P/Q))
// = NTT(a'*skAlice*u + (a'*a*skBob*skAlice - a'*a*sigmaAlice) * (P/Q))
ringQ.AtLevel(plevel).MulCoeffsMontgomery(aprime[i], rhoAlice[i], alpha[i])
ringQ.AtLevel(plevel).INTT(alpha[i], alpha[i])
// alpha = (a'*skAlice*u + (a'*a*skBob*skAlice - a'*a*sigmaAlice) * (P/Q)) * (M/P)
// = a'*skAlice*u*(M/P) + a'*a*skBob*skAlice*(M/Q) - a'*a*sigmaAlice*(M/Q)
ringQ.AtLevel(plevel).DivRoundByLastModulusMany(plevel-mlevel, alpha[i], buff, alpha[i])
alpha[i].Resize(mlevel)
// alpha = - a'*skAlice*u*(M/P) - a'*a*skBob*skAlice*(M/Q) + a'*a*sigmaAlice*(M/Q)
// = - a'*skAlice*u*(M/P) - a'*a*skBob*skAlice*(M/Q) + a'*a*(skBob*skAlice - sigmaBob)*(M/Q)
// = - a'*skAlice*u*(M/P) - a'*a*skBob*skAlice*(M/Q) + a'*a*skBob*skAlice*(M/Q) - a'*a*sigmaBob*(M/Q)
// = - a'*skAlice*u*(M/P) - a'*a*sigmaBob*(M/Q)
ringQ.AtLevel(mlevel).Neg(alpha[i], alpha[i])
}
elapsed = time.Since(start)
fmt.Printf("3.(d) : %s\n", elapsed)
TotalTime += elapsed
AliceTime += elapsed
fmt.Printf("Total time Second Message : %s\n", TotalTime)
fmt.Printf("Alice time Second Message : %s\n", AliceTime)
fmt.Printf("Bob time Second Message : %s\n", BobTime)
// ********* VERIFY CORRECTNESS *********
nerrors = 0
for i := 0; i < n; i++ {
// alpha = - a'*skAlice*u*(M/P) - a'*a*sigmaBob*(M/Q)
// beta = u*v + a'*skAlice*u*(M/P) + a'*a*sigmaBob*(M/Q)
// u * v = alpha + beta
ringQ.AtLevel(mlevel).NTT(v[i], checkMessage1b)
ringQ.AtLevel(mlevel).MulCoeffsMontgomery(u[i], checkMessage1b, checkMessage1a)
ringQ.AtLevel(mlevel).INTT(checkMessage1a, checkMessage1a)
ringQ.AtLevel(mlevel).Add(alpha[i], beta[i], checkMessage1b)
if !ringQ.AtLevel(mlevel).Equal(checkMessage1a, checkMessage1b) {
nerrors++
}
}
fmt.Printf("Errors on Second Message : %d\n", nerrors)
fmt.Printf("\n")
}
}